Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety of dimension $n$.

The polynomials generating the ideal of $V_{n,\, d}$ are well-known: they are minors of a suitable matrix.

Question.Is it possible to cut-out $V_{n, \, d}$ set-theoretically with fewer equations?

For instance, we need three quadrics to generate the ideal of the twisted cubic $V_{1,3}\subset\mathbb{P}^3$, but $V_{1,3}$ is the set-theoretic intersection of a quadric and a cubic.

I am particularly interested in the case $V_{2,\,3}\subset\mathbb{P}^9$.